Spontaneous Symmetry Breaking

BROWN-HET-1555 The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles Gerald S. Guralnik1 Physics Department2. Brown University, Providence — RI. 02912 Abstract I discuss historical material about the beginning of the ideas of spontaneous symmetry breaking and particularly the role of the Guralnik, Hagen Kibble paper in this development. I do so adding a touch of some more modern ideas about the extended solution-space of quantum field theory resulting from the intrinsic nonlinearity of non- trivial interactions. Contents 1 Introduction 1 2 What was front-line theoretical particle physics like in the early 1960s? 2 3 A simplified introduction to the solutions and phases of quantum field theory 3 4 Examples of the Goldstone theorem 7 5 Gauge particles and zero mass 10 arXiv:0907.3466v1 [physics.hist-ph] 20 Jul 2009 6 General proof that the Goldstone theorem for gauge theories does not require physical massless particles 13 7 Explicit symmetry-breaking solution of scalar QED in leading order, show- ing formation of a massive vector meson 15 8 Our initial reactions to the EB and H papers 18 9 Reactions and thoughts after the release of the GHK paper 21 [email protected] 2http://www.het.brown.edu/ 1. Introduction 1 1 Introduction This paper is an extended version of a colloquium that I gave at Washington University in St. Louis during the Fall semester of 2001. But for minor changes, it corresponds to my recent article in ĲMPA [1]. It contains a good deal of historical material about the beginning of the development of the ideas of spontaneous symmetry breaking as well as a touch of some more modern ideas about the extended solution-space (also called vacuum manifold or moduli space) of quantum field theory due to the intrinsic nonlinearity of non-trivial interactions [2, 3]. While I have included considerable technical content, a reader interested only in the historical material should be able to follow the relevant content by just ignoring the equations. The paper is written from a very personal point of view. It puts particular focus on the work that I did with Richard Hagen and Tom Kibble (GHK) [4]. Our group and two others, Englert-Brout (EB) [5] and Higgs (H) [6, 7] worked on what is now known as the “Higgs” phenomenon. I discuss how I understand symmetry breaking (a viewpoint which has changed little in basics over the years) and cover the evolution of our 1964 (GHK) work, which was done in its entirety without any knowledge of others working on the same problem of symmetry breaking and gauge systems. Later in this document, I will make a few brief statements accentuating the differences between our work and that of EB and H. An intense collaborative effort between Hagen, Kibble and me enabled us to produce a paper that to this day gives us great satisfaction. The physics I am about to describe was very exciting. Although we knew the work was basic, we had no appreciation as to how important the concepts involved would become. Despite the current significance of our work, to me the most important thing that has come from it is my enduring long friendship with Dick and Tom. We came together to do this work because of fortunate overlaps of interests and geography. Hagen had been my friend and physics collaborator since our undergraduate days at MIT. I went to Harvard for graduate school, while Hagen stayed at MIT. At that time, in practice, there was little difference in the training a particle theorist received at either of these universities. We could easily take courses at either place and we were constantly doing the short commute between the schools to “cherry pick” course offerings. Many of us from Harvard attended a particle phenomenology course taught by Bernard Feld at MIT. Also, in the mix, was the painful but very important year-long mathematical methods course taught at MIT by Feshbach using his book with Morse. Almost all of us, Harvard or MIT, attended Schwinger’s field theory courses at Harvard. In addition, I took the field theory course at MIT taught by Ken Johnson, who was Hagen’s thesis advisor. This course was beautifully done and more calculational in nature than Schwinger’s courses. I also sat in on Weisskopf’s nuclear physics course at MIT, which was really fun and, in fact, largely taught by Arthur Kerman. Of course, I took most of the “standard” graduate courses at Harvard. Roy Glauber and Wally Gilbert taught courses that were so well reasoned that I find my notes from their classes still useful. Hagen and I wrote our first paper [8] together when we were graduate students, and we continued to talk about life and physics after he went to Rochester as a postdoctoral fellow, while I concluded my thesis on symmetry breaking and spin-one fields with primary emphasis on Lorentz symmetry breaking in four fermion vector-vector couplings [9, 10]. I was working under the direction of Walter Gilbert who, by this time, had largely switched over to biology (Nobel Prize in Chemistry, 1980). Early in 1964, I passed the thesis exam and with Susan, 2. What was front-line theoretical particle physics like in the early 1960s? 2 my new wife, and a NSF postdoctoral fellowship went, in February, to Imperial College (IC) in London, where I immediately met Tom and soon began the discussions that eventually led to our three-way collaboration and the GHK paper. While I recount the history of our work, I will do so embedding a fair amount of physics reflecting how we understood symmetry breaking. Even after all these years, I feel our understanding is still appropriate and indeed recently it helped form the basis of what I believe is a fundamental approach to understanding the full range of solutions (solution-space — this used to be called “vacuum manifold”; however, nowadays it is generally referred to as “moduli space”) of quantum field theories [2, 3]. Some constructs from these modern papers are used in the following discussion to help clarify my points. 2 What was front-line theoretical particle physics like in the early 1960s? To set our work in perspective, it is helpful to review what high energy physics was like when this work was being done. Many “modern” tools such as the Feynman path integral and the now nearly forgotten Schwinger Action Principle were already available. However, the usual starting point for any theoretical discussion was through coupling constant perturbation theory. The power of group theory beyond SU(2) was just beginning to be appreciated, and was realized through flavor SU(3) as introduced by Gell-Mann and Ne’eman. The Ω−, needed to fill in the baryon decuplet (10 particles) was found in 1963. The Gell-Mann–Zweig quark (ace) ideas had just been formulated, but were far from being completely accepted. There was no experimental evidence for quarks, and the ideas about color that allowed three quarks to make a nucleon only began to take form in 1964 [11]. Calculation methods were manual and limited. For the most part, coupling constant perturbation was the only tool available to try to get valid quantitative answers from quantum field theory. Because of the doubts that field theory could ever move beyond perturbation theory, S-Matrix theory was king, at least far west of the Mississippi River. Current algebra, a mixture of symmetry and some dynamics was beginning to take shape with work by Nambu and Lurie [12], but was still in the wings. Indeed, starting in the 1960s, several of the major contributions to what was to become a dramatic reinvigoration of quantum field theory came from the works of Nambu and collaborators. The Nambu–Jona-Lasinio model [13, 14] described by the interaction: ¯ 2 ¯ 2 g (ψψ) − (ψγ5ψ) played a key role in the development and the understanding of spontaneous symmetry breaking in quantum field theory. There is no bare mass term in this interaction and consequently the action has a conserved chiral current. This interaction, in itself, is disturbing (as were the four-fermion interactions used to describe weak processes) because perturbation theory in g produces a series of increasingly primitively divergent terms, making this expansion unrenormalizable. Nambu and Jona-Lasinio studied this model by imposing a constraint requiring that hψψ¯ i= 6 0, which induces a non-vanishing fermion mass, and thus seems in- 3. A simplified introduction to the solutions and phases of quantumfieldtheory 3 consistent with (or “breaks”) chiral symmetry. Based on this assumption, they formulated a new leading order approximation (not a coupling constant perturbation theory). This approximation is, in fact, consistent with the conservation of the chiral current, thereby as- suring that, despite the induced mass, the approximation remains consistent with the basic requirements of the field equations for this action. In addition, their results showed that there was a zero mass composite particle excited by (ψγ¯ 5ψ). This particle is now called a Goldstone boson or occasionally a Nambu-Goldstone boson. The name was acquired after developments in other papers [15, 16]. Stated somewhat more generally, Ref [16] proves that if the commutator of a conserved charge (resulting from a continuous current) with a local field operator has non-vanishing vacuum expectation value, then the local field operator must have a massless particle in its spectrum. This result is exact and not confined to a leading order approximation. This is the Goldstone theorem and it is always true, provided its basic assumptions, outlined carefully in what follows, are satisfied.

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